A New Technique to Solve the Instant Insanity Problem

Instant Insanity [1] consists of four cubes, each of whose six faces are colored with one of the four colors: red, blue, white, and green. The object is to stack the cubes in such a way that each of the four colors appears on each side of the resulting column. See figure 1 below[2]. Traditionally, this could be solved using graph theory.However, in this article, we introduce a new technique to solve the problem without using graph theory. We also used a Perl programming language to implement the new approach for the Instant Insanity

The basis number of the powers of the complete graph

A basis of the cycle space C(G) of a graph G is h-fold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that C(G) has an h-fold basis. MacLane [3] showed that a graph G is planar if and only if b(G) ≤ 2. Schmeichel [4] proved that b(Kn) ≤ 3, and Banks and Schmeichel [2] proved that b(K d 2) ≤ 4 where Kd 2 is the d-dimesional hypercube. We show that b(K d n) ≤ 9 for any n and d, where K d n is the cartesian d-th power of the complete graph Kn. 0 Keywords: cycle space of a graph, basis number, powers of complete graphs

An upper bound on the basis number of the powers of the complete graphs

summary:The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$

New edge neighborhood graphs

summary:Let $G$ be an undirected simple connected graph, and $e=uv$ be an edge of $G$. Let $N_G(e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$. Let $\mathcal N_e$ be the class of all graphs $H$ such that, for some graph $G$, $N_G(e)\cong H$ for every edge $e$ of $G$. Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $\mathcal N_e$. Balasubramanian and Alsardary [1] obtained some other graphs in $\mathcal N_e$. In this paper we given some new graphs in $\mathcal N_e$